The present invention relates to a new and distinct variety of white peach tree, Prunus persica (L.) Batsch, which has been given the variety denomination ‘SWEETEMBER’. This new tree produces fruits with a long shelf life without alteration both on the tree after growth completion and after harvesting, very good eating quality, clingstone white and slightly green flesh fruits for fresh market in September in the Pyrénées-Orientales department, France. Contrast is made to ‘MAILLARDIVA’ white peach tree (non patented), for reliable description. ‘SWEETEMBER’ is a promising candidate for commercial success in that it has very attractive fruits with very long shelf life without alteration both before and after harvesting.